If you care about Neandertal behavior and haven't read this 2004 article by John Speth, you really should treat yourself: "News flash: Negative evidence convicts Neanderthals of gross mental incompetence"
Neanderthals didnt make blades, or at least not as often, or maybe not as well, as moderns did (they didnt make microliths either, or stainless steel for that matter, but they did make great triangles; unfortunately rectangles are in these days, not triangles). They didnt carve bone or ivory (nor did they work fiberglass, though they may have carved a lot of wood, judging by recent use-wear evidence). They didnt paint the walls of their caves, despite ample opportunity to do so (but, then, painting cave walls, even in the Upper Paleolithic, was truly the exception, not the rule). They did have spears, we have some of them (this, no doubt, is what all of their woodworking was about), but try as they might they just couldnt throw them (the points that somehow managed to find their way through hair and hide of sizeable prey to become squarely embedded in solid bone notwithstanding). They had no large formal fireplaces (whats wrong with some of the hearths at Kebara?), so they couldnt sit cozily face-to-face around the fire at night holding hands, roasting marshmallows and singing campfire songs, hence they must have lacked true language (and Girl Scouts; in fact, the conspicuous absence of marshmallows throughout the Upper Paleolithic clearly testifies to the lack of language as we know it until very late in the Holocene).
If you haven't been following paleoanthropology for long, you may find it difficult to believe that Serious Scientists have proposed some of the nonsense that Speth skewers (for example, that Neandertals lived bison-like in cow-calf and bull groups who only joined when rutting). Speth's tour through bad ideas is a ribald pleasure.
Yet there is an important point in the paper, which is why I'm citing it today: The material record of Neandertals is in many respects within the range of hunter-gatherers who are unquestionably modern humans:
What is the relevance of the North American record to the question of Neanderthals mental hardware? Simply this: most sites that date to the Paleo-Indian and early Archaic periods periods that together last some 5,000 to 6,000 years and represent nearly half of the known occupation span of the New World have little or no evident internal structure (unless you want to count patches or scatters as structure; the rare high-resolution examples share the same humdrum carnivore dichotomy that was used to torpedo the poor Neanderthal); there are few if any formal hearths (isolated patches and lenses of ash are much more the norm); burials are extremely rare or absent altogether, and those few that exist have little or nothing in the way of ornaments or grave accompaniments; huts are generally absent or very controversial; and art of any non-perishable sort is virtually non-existent (they certainly didnt paint cave walls; in fact, we are hard put in most cases to find anything that even remotely smacks of symbolism). If we were to use the same criteria that we apply to Neanderthals, we would have to conclude that the inhabitants of North America up until only a few thousand years ago were to put it in politically correct terms cognitively challenged. The parallels with the record of the Middle Paleolithic are even more striking if we exclude from consideration the few dry caves in western North America and waterlogged sites in Florida of late Paleo-Indian and early Archaic age which have miraculously preserved tantalizing traces of perishable basketry, textiles and other unusual items.
Then around 5,000 years ago, give or take a millennium, came North Americas counterpart to Eurasias Upper Paleolithic revolution. We suddenly see an explosion of art intricately shaped or carved and sometimes engraved exotica of shell, antler, bone, stone, tortoise shell and native copper, including cups, tubes, pendants, beads, pins, rattles, atlatl hooks, bannerstones and gorgets; there are also remnants of probable medicine bags, traces of textiles, decorated baskets and ubiquitous red ochre the whole nine yards. Many of the raw materials came from distant lands marine shell from the Gulf of Mexico, sharks teeth from the mid-Atlantic states, copper from Lake Superior, galena and mica from Illinois and the Appalachians. This is also the time when we begin to see burials clustered together in real cemeteries, not just peppered here and there over the archaeological landscape; and many of these burials are elaborately decked out with ornaments and other exotica, so much so at times that we begin to speculate about the beginnings of prestige enhancement and wealth display, about big men, about reduced mobility and increasing conflict, about the growing importance of inter-group exchange and political alliances, about the very seeds of societal inequality and hierarchy. This is the bread and butter of North American archaeology. And, while there is lots to disagree and argue about (particularly about what is cause and what is effect), all seem to agree that in some form or other what we are seeing over the course of the Archaic is the playing out of gradually increasing populations that were slowly filling in the landscape, reducing peoples ability to vote with their feet when things got tough, and thereby compelling them to begin playing with alternative economic, social and political strategies for maintaining the delicate balance between war and peace in a word, social, technological, economic and political intensification. No one, of course, would believe for a nanosecond that in the artless and styleless silence of the early Archaic we are dealing with a cognitively impaired proto-human.
The Paleo-Indian/Archaic period transition is obviously not a perfect analogy for the Upper Paleolithic transition in Europe. Speth himself is explicit that this example does not prove anything about Neandertals.
But it does illustrate a double-standard. Recent archaeological peoples are prima facie "modern" in behavior without showing evidence for "symbolic" interactions. With Paleo-Indian sites, the subsistence strategy itself argues for a complex logistical organization, even though habitation sites, kill sites and artifacts comport with the lack of structural complexity of many hunter-gatherer groups in the ethnographic present. Some scientists have presented similar arguments for Neandertals.
Personally, I think that "cognitive modernity" is a red herring. Today's people learn some kinds of technical and symbolic complexity that were never present in ancient peoples. Some people living today in Western cultures, despite all our educational efforts, fail to attain levels of technical knowledge that are regular outcomes for the majority of people in the same environment. Human performance varies continuously.
I assert that it is unreasonable to suppose that Neandertals had a "stupid gene". If so, it should be just as unreasonable to suppose that a "smart gene" could explain the evolution of human cognition during the last 100,000 years. These unrealistic assumptions are widespread, and impede our understanding just as thoroughly as assumptions about the nature of biological species impeded our understanding of Neandertal ancestry of living human populations. Some archaeologists have concluded that Neandertal cognition is an either/or proposition. Some look at Neandertals, find a lack of evidence that they behave identically to later people, and conclude that the Neandertals were therefore unquestionably cognitive inferiors. Others look at Neandertals, find some signs of modern-like behaviors, and conclude that Neandertals were therefore unquestionably our cognitive equals.
Cognition in modern humans varies continuously across many axes of variation. No two humans are cognitively identical in outcomes. Nor can we appeal to "cognitive capacity", a meaningless abstraction unless we are discussing a particular structured learning environment in which the outcomes are potentially measurable. Will we someday raise a Neandertal in a human society to see whether and how they attain the skills and abilities we consider essential?
I suspect somewhere within the broad scope of human variation in learning, we already are.